Pie Chart Analysis: Calculating Percentages and Angles

Task A

To find the percentage of people whose favorite swimming stroke is freestyle, we need to determine what portion of the pie chart represents freestyle.

Calculate the angle for Sidestroke:

The pie chart must represent 360 degrees in total.
We know the angles for Freestyle (162 degrees) and Backstroke needs to be calculated from the percentage.
First, calculate the angle for Backstroke: 20% of 360 degrees.

Calculate the angle for Backstroke:

Angle = 20% * 360 degrees = 0.20 * 360 = 72 degrees

Calculate the angle for Sidestroke:

Angle = 360 - 162 - 72 = 126 degrees

Calculate the percentage for Freestyle:

Percentage = (Freestyle angle / 360) * 100
Percentage = (162 / 360) * 100 = 45%

Answer: 45% of people said their favorite was freestyle.

Task 2A

We need to identify which of the statements are true based on the bar graph showing the number of electronic devices sold.

The largest total number of electronic devices sold was on Wednesday:

On Wednesday: TV (20) + Phone (30) + Tablet (25) = 75
On Monday: TV (20) + Phone (15) + Tablet (10) = 45
On Tuesday: TV (20) + Phone (30) + Tablet (15) = 65
The largest number of devices was indeed sold on Wednesday (75).
TRUE

The smallest number of TVs sold was on Monday:

TV sales were the same (20) on all three days.
FALSE

The same number of phones was sold on each of Monday, Tuesday and Wednesday:

Monday: 15 Phones
Tuesday: 30 Phones
Wednesday: 30 Phones
FALSE

On Tuesday, more tablets were sold than any other type of electronic device:

On Tuesday: TV (20), Phone (30), Tablet (15)
The statement is false because more phones were sold than tablets on Tuesday.
FALSE

Answer: The only true statement is: "The largest total number of electronic devices sold was on Wednesday."

## Task 4A To find the size of angle $x$, we need to understand the properties of regular polygons, specifically pentagons and hexagons. 1. **Interior Angle of a Regular Pentagon:** * The formula for the interior angle of a regular polygon is: $\frac{(n-2) \times 180}{n}$, where $n$ is the number of sides. * For a pentagon, $n = 5$. * Interior angle = $\frac{(5-2) \times 180}{5} = \frac{3 \times 180}{5} = \frac{540}{5} = 108$ degrees. 2. **Interior Angle of a Regular Hexagon:** * For a hexagon, $n = 6$. * Interior angle = $\frac{(6-2) \times 180}{6} = \frac{4 \times 180}{6} = \frac{720}{6} = 120$ degrees. 3. **Calculating Angle $x$:** * Angle $x$ is formed by the combination of one interior angle from the pentagon and two interior angles from the hexagons. * $x = 360 - (\text{Interior angle of pentagon} + 2 \times \text{Interior angle of hexagon})$ * $x = 360 - (108 + 2 \times 120)$ * $x = 360 - (108 + 240)$ * $x = 360 - 348$ * $x = 12$ degrees **Answer:** The size of angle $x$ is 12 degrees.

## Task 5A To find the size of angle $d$, we need to use the properties of triangles and polygons. 1. **Identify the Triangle:** * We have a triangle with angles $66^\circ$, $76^\circ$, and an unknown angle. 2. **Find the Unknown Angle in the Triangle:** * The sum of angles in a triangle is $180^\circ$. * Unknown angle = $180^\circ - 66^\circ - 76^\circ = 180^\circ - 142^\circ = 38^\circ$ 3. **Identify the Pentagon:** * We have an irregular pentagon with angles $112^\circ$, $23^\circ$, $102^\circ$, $76^\circ + d$, and $66^\circ + 38^\circ$. 4. **Find the Sum of Interior Angles of a Pentagon:** * The sum of interior angles of a polygon with $n$ sides is $(n-2) \times 180^\circ$. * For a pentagon, $n = 5$. * Sum of interior angles = $(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$ 5. **Set Up the Equation:** * $112^\circ + 23^\circ + 102^\circ + (76^\circ + d) + (66^\circ + 38^\circ) = 540^\circ$ 6. **Solve for $d$:** * $112 + 23 + 102 + 76 + d + 66 + 38 = 540$ * $417 + d = 540$ * $d = 540 - 417$ * $d = 123^\circ$ **Answer:** The size of angle $d$ is 123 degrees.

## Task A To find the percentage of people whose favorite swimming stroke is freestyle, we need to determine what portion of the pie chart represents freestyle. 1. **Identify known angles/percentages:** * Sidestroke angle = 162 degrees (pink sector) * Backstroke percentage = 20% (orange sector) 2. **Calculate the angle for Backstroke:** * Angle = 20% of 360 degrees * Angle = 0.20 * 360 = 72 degrees 3. **Calculate the angle for Freestyle:** * The total angle in a circle is 360 degrees. * We know Sidestroke (162 degrees) and Backstroke (72 degrees). * Freestyle angle = 360 - (Sidestroke angle + Backstroke angle) * Freestyle angle = 360 - (162 + 72) * Freestyle angle = 360 - 234 * Freestyle angle = 126 degrees 4. **Calculate the percentage for Freestyle:** * Percentage = (Freestyle angle / 360) * 100 * Percentage = (126 / 360) * 100 * Percentage = 0.35 * 100 = 35% **Answer:** 35% of people said their favorite was freestyle. ## Task B We need to find the central angle of the backstroke sector. * The problem states that 20% of people said their favorite was backstroke. * To convert a percentage to an angle in a pie chart, multiply the percentage (as a decimal) by 360 degrees. * Angle = 20% * 360 degrees * Angle = 0.20 * 360 * Angle = 72 degrees **Answer:** The central angle of the backstroke sector should be 72 degrees.

## Task 2A We need to identify which of the statements are true based on the bar graph showing the number of electronic devices sold. 1. **The largest total number of electronic devices sold was on Wednesday:** * On Monday: TV (20) + Phone (15) + Tablet (10) = 45 * On Tuesday: TV (20) + Phone (30) + Tablet (15) = 65 * On Wednesday: TV (20) + Phone (30) + Tablet (25) = 75 * Comparing the totals: 45 (Monday), 65 (Tuesday), 75 (Wednesday). * The largest number of devices was indeed sold on Wednesday (75). * **TRUE** 2. **The smallest number of TVs sold was on Monday:** * TV sales (blue bars) were 20 on Monday, 20 on Tuesday, and 20 on Wednesday. * The number of TVs sold was the same on all three days, not smallest on Monday. * **FALSE** 3. **The same number of phones was sold on each of Monday, Tuesday and Wednesday:** * Phone sales (green bars): * Monday: 15 Phones * Tuesday: 30 Phones * Wednesday: 30 Phones * The number of phones sold was not the same on Monday compared to Tuesday and Wednesday. * **FALSE** 4. **On Tuesday, more tablets were sold than any other type of electronic device:** * On Tuesday: * TV: 20 * Phone: 30 * Tablet: 15 * On Tuesday, 30 phones were sold, which is more than the 15 tablets sold. * **FALSE** **Answer:** The only true statement is: "The largest total number of electronic devices sold was on Wednesday."

## Task 4A To find the size of angle $x$, we need to understand the properties of regular polygons, specifically pentagons and hexagons. 1. **Interior Angle of a Regular Pentagon:** * The formula for the interior angle of a regular polygon is: $\frac{(n-2) \times 180^\circ}{n}$, where $n$ is the number of sides. * For a pentagon, $n = 5$. * Interior angle of pentagon = $\frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ$. 2. **Interior Angle of a Regular Hexagon:** * For a hexagon, $n = 6$. * Interior angle of hexagon = $\frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ$. 3. **Calculating Angle $x$:** * Angle $x$ is formed at a point where one pentagon and two hexagons meet. The sum of angles around a point is $360^\circ$. * The angles meeting at that point are: * One interior angle of the pentagon. * One interior angle of the first hexagon. * One interior angle of the second hexagon. * Angle $x$. * So, $x + (\text{Interior angle of pentagon}) + (\text{Interior angle of hexagon}) + (\text{Interior angle of hexagon}) = 360^\circ$. * $x + 108^\circ + 120^\circ + 120^\circ = 360^\circ$ * $x + 348^\circ = 360^\circ$ * $x = 360^\circ - 348^\circ$ * $x = 12^\circ$ **Answer:** The size of angle $x$ is 12 degrees.

## Task 5A To find the size of angle $d$, we need to use the properties of triangles and polygons. 1. **Find the missing angle in the triangle:** * The triangle has angles $66^\circ$ and $76^\circ$. * The sum of angles in a triangle is $180^\circ$. * The third angle in the triangle (let's call it $A$) is $180^\circ - 66^\circ - 76^\circ = 180^\circ - 142^\circ = 38^\circ$. 2. **Identify the vertices and angles of the irregular pentagon:** * The pentagon has 5 sides and 5 interior angles. Let's list them: * Angle 1: $112^\circ$ * Angle 2: $23^\circ$ * Angle 3: $102^\circ$ * Angle 4: This angle is composed of $76^\circ$ and $d$. So, it is $(76^\circ + d)$. * Angle 5: This angle is composed of $66^\circ$ and the angle $A$ we found in step 1 ($38^\circ$). So, it is $(66^\circ + 38^\circ) = 104^\circ$. 3. **Calculate the sum of interior angles of a pentagon:** * The formula for the sum of interior angles of a polygon with $n$ sides is $(n-2) \times 180^\circ$. * For a pentagon, $n = 5$. * Sum of interior angles = $(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$. 4. **Set up and solve the equation for $d$:** * The sum of all interior angles of the pentagon must equal $540^\circ$. * $112^\circ + 23^\circ + 102^\circ + (76^\circ + d) + (66^\circ + 38^\circ) = 540^\circ$ * $112 + 23 + 102 + 76 + d + 104 = 540$ * Combine the known numerical values: $112 + 23 + 102 + 76 + 104 = 417$ * So, $417^\circ + d = 540^\circ$ * $d = 540^\circ - 417^\circ$ * $d = 123^\circ$ **Answer:** The size of angle $d$ is 123 degrees.